A more concrete definition

If the field F has characteristic zero we have the following more
concrete definition. Let R:=F⁢[{Xi}i∈I] be the polynomial
ring over F in indeterminatesXi labeled by I. For any i∈I, let ∂i denote the partial differential operator with
respect to Xi. Then the |I|-th Weyl algebra is the set W of all
differential operators of the form

The equivalence of these definitions can be seen by replacing the
generatorsQi with left multiplication by the indeterminates Xi,
the generators Pi with the partial differential operator
∂i, and the tensor product with operator multiplication, and
observing that ∂i⁡Xj-Xj⁢∂i=δi⁢j. If, however,
the characteristicp of F is positive, the resulting homomorphism
to W is not injective, since for example the expressions
∂ip and Xin commute, while Pi⊗p and
Qi⊗n do not.